Question

factor completely

3x^2-14x-24


I do not see the gcf for the problem
I am totally lost can some explain please I have several these problems and i need to figure out how to solve them

You put

3x^2-14x-24 = (a x + b)(c x + d)

Equating the coefficients on both ides gives:

3 = a c

This means that we can take
a = 3 and c = 1:

3x^2-14x-24 = (3 x + b)(x + d)

Check that you don't get anything new if you take a = -3 and c = -1 or if you interchange a and c.

Equating the coefficient of x on both sides gives:

3 d + b = -14
Equating the constant term on both sides gives:

b d = -24

Some trial and error yields:

b=4 and d = -6 d, so we obtain:

3x^2-14x-24 = (3 x + 4)(x -6)


When factoring using this method you use trial and error. The factorization of the constant term, in this case -24 plays an important role. What you are using is that since the factorization has to be true for all x, it has to be true also for x = 0.

This means that you can often simply matters considerably by considering some other point instead of x = 0. In this case, if you take x = -1 then the function is -7 which has a unique factorization (up to signs)

So, let's see how that works out in this case. Let's start here:

3x^2-14x-24 = (3 x + b)(x + d)

If you substitute x = 0 in both sides then you get the equation:

-24 = b d

But there are many possibilities here. So, let's take x = -1 instead. We then get:

-7 = (b - 3)(d - 1)

Possibility 1:

b-3 = -7 and d-1 = 1

doesn't work because then b = -4 and d = 2 and b d isn't equal to 24.

Possibility 2:

b-3 = 7 and d-1 = -1
a non-starter because then d becomes zero and the product b d = 0.


Possibility 3:

b-3 = 1 and d-1 = -7

This is a possible solution because you get b = 4 and d = 6 and b d = 24.

You still need to check that

3 d + b = -14

when solving the problem in this way.

Typo:

You put

3x^2-14x-24 = (a x + b)(c x + d)

Equating the coefficients of x^2 on both sides gives:

3 = a c

A few more typos in the last part of the solution:

Possibility 1:

b-3 = -7 and d-1 = 1

doesn't work because then b = -4 and d = 2 and b d isn't equal to -24.

Possibility 2:

b-3 = 7 and d-1 = -1
a non-starter because then d becomes zero and the product b d = 0.


Possibility 3:

b-3 = 1 and d-1 = -7

This is a possible solution because you get b = 4 and d = -6 and b d = -24.